Continuous functions inner product definition

Question

If we have two functions F1(x) & F2(x) defined on [x1,x2] then their inner product is $\int_{x_1}^{x_2} F_1(x)F_2(x)dx$ The origin of the definition is that F1(x) can be represented as a vector of infinite dimensions each containing F1(x) (the value of F1) at this dimension, so F1(x).F2(x)=$\sum_{x_1}^{x_2} f_{1i} f_{2i} $, so where did the "dx" come from in the integral $\int F_1(x)F_2(x)dx$? what I know is that $\int_{x_1}^{x_2} F_1(x)F_2(x)dx$=$\sum_{x_1}^{x_2} f_{1i} f_{2i} \Delta x$ when $\Delta x \rightarrow\infty$ so how come we convert the product to an integral when the summation doesn't have $\Delta x$ ??